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ASYMPTOTIC BEHAVIOUR OF THE STOCHASTIC MAKI–THOMPSON MODEL WITH A FORGETTING MECHANISM ON OPEN POPULATIONS

Part of: Stochastic analysis

Published online by Cambridge University Press:  06 November 2020

HAIJIAO LI Open the ORCID record for HAIJIAO LI [Opens in a new window]  and
KUAN YANG Open the ORCID record for KUAN YANG [Opens in a new window]
HAIJIAO LI
Affiliation:
School of Business Administration, Hunan University, Hunan, China; e-mail: [email protected] and [email protected].
KUAN YANG
Affiliation:
School of Business Administration, Hunan University, Hunan, China; e-mail: [email protected] and [email protected].
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Abstract

Rumours have become part of our daily lives, and their spread has a negative impact on a variety of human affairs. Therefore, how to control the spread of rumours is an important topic. In this paper, we extend the classic Maki–Thompson model from a deterministic framework to a stochastic framework with a forgetting mechanism, because real-world person-to-person communications are inevitably affected by random factors. By constructing suitable stochastic Lyapunov functions, we show that the asymptotic behaviour of the stochastic rumour model is governed by the basic reproductive number. If this number is less than one, then the solution of the stochastic rumour model oscillates around the rumour-free equilibrium under extra mild conditions, indicating the extinction of the rumour with a probability of one. Otherwise, the solution always fluctuates around the endemic equilibrium under certain parametric restrictions, implying that the rumour will continually persist. In addition, we discuss a possible intervention strategy that stops the spread of rumours by strengthening the intensity of white noise, which is very different from the deterministic rumour model without white noise. Also, numerical simulations are conducted to support our analytical results.

Keywords

extinctionstationary distributionstochastically asymptotically stablestochastic rumour model

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60H40: White noise theory
Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 2 , April 2020 , pp. 185 - 208
Copyright
© Australian Mathematical Society 2020

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